By
Robin Johnson
Unpublished manuscript, 2001.
This paper examines the constant elasticity production (C.E.S.) function as an appropriate vehicle to analyse rates of return on the factors of production and technology growth. The objective was to find a satisfactory theoretical formulation that not only fitted aggregate sector data but also identified the role of research and development (Research and Development) investment in the production process. The formulation is demanding of the data and if one part of the fitted equation is non-consistent with the theory, then measured rates of return and rates of technology growth may be difficult to identify. As with previous studies by the author, when R & D investment is measured as a stock of accumulated capital available to producers at the beginning of the year, the rate of return is negative on a year-to-year basis. This result indicates that the accumulated stock of knowledge as measured by a depreciated inventory is not having any immediate effects on annual net returns as measured at the aggregate GDP sector level.
This article is concerned with matching econometric analysis to textbook production models as exemplified by production functions. The work arose out a series of analyses measuring the rate of return to Research and Development in 10 sectors of the NZ economy (Johnson 1999, 2000). These analyses employed Cobb-Douglas models of production treating R&D as an additional capital input. Critics and reviewers were concerned with the poor performance of the Cobb-Douglas specification of the production process and its general suitability for the task in hand. Editors and reviewers suggested that it might be more important to test the production model employed than recommend the “spurious” rates of return to R&D that were produced with the simpler versions of production theory.
This article therefore starts from a more generalised production function potentially applicable to the agricultural sector. The model employed is the constant elasticity of substitution (CES) production function, of which Cobb-Douglas is a special case i.e. where the elasticity of substitution between labour and capital is equal to unity. There is also a proportional elasticity of substitution model (PES) which incorporates three factors of production that can be employed to verify if the elasticities of substitution vary from unity as well (Solow 1967).
This article therefore is based on a series of experiments with aggregate data for the NZ economy from the Philpott collection testing for the appropriate models which marry production theory with data availability and errors therein (Philpott 1994, 1999). Tests are carried out for the agricultural sector and the appropriate formulation determined. It turns out that aggregate data does not satisfactorily represent enough of the range of response in factor employment to real price changes which would enable the production parameters to be specified more accurately. It is possible to isolate conditions of biassed efficiency growth associated with factors employed without clear guidelines on rates of substitution. The models developed do not provide any new guidance on the rate of return on Research and Development expenditure by firms and government in NZ.
The CES production function can be written (Y is net output):
| (1) | Y = A [δ L p + (1 - d)K p] 1/p |
where
p = 1 - 1/ σ = the substitution parameter
A = scale parameter, A > 0
δ = distribution parameter, 0 < δ < 1
and σ = 1 / 1- p = elasticity of substitution.
In the CES formulation, the elasticity of substitution is not constrained to unity, and with suitable manipulation of (1) it can be estimated from production data on firms or time series.
It is also recognised that the elasticity of substitution will vary in time as the relative efficiency of the different inputs changes. In order to counteract such biassed efficiency growth in production conditions over time, (1) needs to be re-specified to take account of the respective growth rates in factor efficiency. In the case of the labour input, EL represents the exponential growth in labour efficiency and will take the value in any year:
| (2) | EL(t) = EL(o) e λL t |
where λ L = growth rate of labour efficiency.
Changes in factor efficiency are interpreted as labour augmenting or capital augmenting technological advances by the respective growth paths ofEL and EK. The production function can then be normalised for differential factor efficiency by writing (1) as follows:
| (3) | Y = A [δ(EL.L)p + (1 - δ)(EK.K)p]1/p |
In effect, the labour and capital inputs are expressed in efficiency units which will have an appropriate substitution relationship. The elasticity of substitution is then expressed as the measure of the relationship between factors measured in efficiency units and price changes, thus:
| (4) | σ = | d(EKK / ELL) / (EKK / ELL)
d(w/r)/(w/r) |
= | 1 1 - p |
where w and r are the real price of labour and capital respectively. Without this conversion the elasticity would be measured at the points A0 - A2 in Figure 1a. The use of inputs in efficiency units normalises these isoquants as in Figure 1b and gives a consistent estimate of σ.
It also makes sense at this stage to introduce an adjustment process into the specification that allows a distinction to be made between long-run and short run factor adjustment and output. We could make the supposition that output per labour unit, Y/L, does not respond fully in a given time period to the full adjustment that changes in factor prices would indicate. In the following specification, the change from one year to the next in Y/L is an exponentially weighted average, φ, of the ratio of potential productivity, Y/L*, to past productivity:
| (5) | ln Y/L - ln Y/L-1 = φ (ln Y/L* - ln Y/L-1) 0 < φ < 1 |
For estimation purposes, we first derive the marginal product side relationships in (3) assuming constant returns to scale (Intriligator 1978, pp.273-6). If (3) is differentiated with respect to L, and re-arranged, the following expression for the real wage is obtained:
| (6) | w = ELp ( Y/L ) 1-p |
from which
| (7) | Y/L = w1/1-p EL -p/1-p |
This can be expressed in logarithms for estimation purposes as:
| (8) | ln Y/L = (1/1-p) ln w + [ 1- ( 1/1-p)] ln EL |
Since EL has no absolute value, we need to substitute (2) in place of this term to get:
| (9) | ln Y/L = a + s ln w + (1- s)lL t |
where a = (1- s ) ln EL(o)
and s = 1/1-p
Finally we can incorporate the adjustment term, f , into the final estimating equation for labour[capital] productivity :
| (10) | ln Y/L = φ a + φ σ ln w + σ λL (1- σ) t + (1 - σ) ln Y/L-1 |
where
est a* = φa = φ(1 - σ) ln EL(o)
est b1 = φσ
est b2 = φλ(1 - σ)
est b3 = (1 - σ)
If (7) is inverted and multiplied by w, we obtain an expression for the labour factor shares:
| (11) | wL /Y = w -p/1-p EL p/1-p |
which in turn gives the following estimation equation:
| (12) | ln FL = φ a + φ (1 - s ) ln w + φ λL ( σ - 1) t + (1 - φ ) ln FL -1 |
This equation gives an alternative estimation for σ12 and efficiency growth. It applies also to the capital share equation if the cost of capital is exogenous.
In certain circumstances it may be preferable to investigate a 3-variable production function which will take a different form to the CES function. Consider a function with a third input category, M, and a gross output dependent variable, G. Each substitution parameter will be different from the general substitution parameter thus:
| (13) | G = [(ELL) p1 + (EKK) p2 + (EMM) p3 ] 1/p |
This function, developed by Mukerjee (1963) is called a proportional elasticity of substitution model (PES) because it preserves a formal relationship between the elasticities of substitution and the individual substitution parameters, viz
| (14) | σ12 σ13 |
= | (1 - p3) (1 - p2) |
When (13) is differentiated with respect to L, the expression for the marginal product of labour becomes:
| (15) | dG dL |
= | P1 P |
(1 - p3) (1 - p2) |
ELp1 = w |
which gives the following equation for estimation:
| (17) | σ 12 = | 1 | - | 1 | ||
| Fl + | Fk + | Fm | ( 1 - p1) ( 1 - p2) | |||
| (1 - p1) | (1 - p2) | (1 - p3) | ||||
where F is the factor share for each input category.
This approach might be useful where the deduction of intermediate inputs from gross output to obtain net output creates efficiency bias in the labour and capital inputs. It is plausible that intermediate inputs are a vehicle for the introduction of some of the gains from technical efficiency.
All the data for the estimation is derived from information in the Philpott estate. The period is 1962-1998. Each variable is defined in turn:
Y = net output in real terms (1982-83 prices) for each sector;
K = gross capital stock on a vintage basis at 1982-83 prices;
L = labour employed in fulltime equivalents;
M = intermediate inputs at 1982-83 prices;
G = gross output at 1982-83 prices (G - M = Y);<//p>
W = unit compensation of employees in nominal terms;
P = implicit price index for net output;
w = W/P = real wage per labour unit;
m = interest paid on new mortgages;
r = m/P = real interest rate (cost of capital);
t = time trend
s = relative price of intermediate inputs;
p = substitution parameter;
σ = 1/1 - p = elasticity of substitution;
λ = growth rate of factor efficiency
φ = average adjustment rate in dependent variable.
I shall examine each attribute in turn. It is logical to look at the adjustment factor first, then the derivation of the elasticity of substitution and finally the estimate of factor efficiency growth. The sector is the agricultural sector as defined by Statistics NZ. The function is CES.
The adjustment parameter, φ, is the proportion of “true” output that is reached on average in a given year (Table 1). The hypothesis is that there is a lag in reaction to changes in factor conditions expressed through the output variable [prices or physical input]. The estimated coefficients are nearly all significant at the 5% level. Compared with earlier analysis (Johnson 1972) the adjustment process has slowed in the period 1962-98.
The substitution elasticity depends on the response of factors and output to changes in real prices. In some cases (Y/L, Y/K, and Fk), the sign of the estimated coefficient is consistent with the production logic, but in others it is not. The sign of the elasticity of substitution parameter is then pre-determined and also may be inconsistent. In general, the derived substitution elasticities are very low and indicate that there is a lack of flexibility in input substitution. The implication is that farm producers do not easily change resource use as price relativities change between labour and capital. Clearly, this analysis does not support the nice isoquant curvature between two inputs displayed by the CD model. There is a negative substitution elasticity where the capital/labour ratio is the dependent variable in one case. These coefficients do not reach the 5% level of significance so could safely be assumed to be zero. Thus the capital/labour ratio appears to be invariant to respective real prices, even allowing for the ratio to adjust over a period of years. Apparently, the movement in the use of the respective inputs, capital and labour as defined, in relation to the respective factor shares of the same inputs, is perverse in the case of case of labour but consistent in the case of capital.
The coefficient, λ, measures the rate of technological growth associated with each factor input. Labour technological growth exceeds capital technological growth. These results are interpreted to mean that technology is more labour augmenting that capital augmenting. In production theory, this means that the isoquant moves out from the axis in a biassed way (i.e. it slopes more in the direction of the L axis) and as efficiency increases the value of the elasticity between the inputs will change. [in the empirical analysis, labour and capital are measured in efficiency units to overcome this problem]. In Table 1, it is clear that technological growth is labour augmenting though in the period 1962-98 the difference between the two l in (3) and (4) does not pick up the difference in the two growth rates accurately.
| Dependent Variable | Price Parameter | Adjustment Parameter | Substitution Coefficient | Elasticity of Substitution | Technological Growth Parameter |
| Y/L: w | + | 0.54 ** | 0.12 ns | 0.22 | 0.028 ** (λL) |
| Y/K: r | + | 0.35 ** | 0.08 * | 0.25 | 0.013 ns (λ |
| K/L: w/r | - | 0.29 ** | -0.01 ns | -0.05 | 0.009 ** (λL - λK) |
| FL: w | - | 0.18 ** | -0.07 ns | 1.42 | 0.029 ns (λL) |
| FK : r | + | 0.76 ** | 0.33 ** | 0.56 | 0.020 ** (λK) |
| Dependent Variable | Price Parameter | Adjustment Parameter | Substitution Coefficient | Elasticity of Substitution | Technological Growth Parameter |
| G:L,w | + ns | 0.46 ** | 0.06 ns | s12 = -0.14 | 0.016 ** (λL) |
| G:K, r | + ns | 0.38 ** | 0.015 ns | s23 = -0.20 | 0.019 ** (λK) |
| G:M,s | + ns | 0.57 ** | 0.012 ns | s13 = -0.14 | 0.023 ** (λM) |
φ, the adjustment parameter can be regarded as the percentage of true equilibrium output that is achieved on average over the sample of years. As the dependent variable is the same for all 3 equations in Table 2, the variation in f for the period 1962-98 must be caused by the influence of the independent variables used in each equation. In the labour and capital equations the adjustment is about 40-50% per year. But the coefficient for non-factor inputs at 0.57 suggests the response of production to intermediate input change is greater.
Changes in real costs of inputs are associated with increases in total output possibly due to delayed responses in production conditions. Positive changes in the price of non-factor inputs are also associated with increases in gross output as inputs are conserved. The substitution elasticities are all negative and non-significant. These negative values arise from negative relationships between gross output and labour and capital inputs. In the adjustment model.+ve change in gross output takes place in years of decline in the inputs.
An estimation of technological growth for labour and capital is still possible, but there is a negative growth coefficient for λM. This suggests that intermediate inputs do not carry augmented technological growth.
In Table 3 we present the results of a 3 factor PES production function analysis where we split off from total capital employed that part of capital which represents the invested capital in Research and Development in agriculture. In fact, of course, the R&D capital was not represented in total capital previously as total capital is based on physical capital in land improvements and equipment. The stock of R&D capital is built up from a base by adding each year's net investment in R&D and subtracting an estimated rate of depreciation [the perpetual inventory method]. Because the capital input is split the dependent variable reverts back to real net output. The rate of depreciation is assumed to be 5% per annum and the stock starting value is capitalised at the first years annual input. As with the general capital stock, the R&D stock is that stock available at the beginning of the production year. The methodology is given in equations (15) and (16).
| Dependent Variable | Price Variable | Adjustment Parameter | Research and Development Coefficient | Technological Growth Parameter |
| Y: L, w | +** | 0.76 ** | - | 0.021 ** (λL) |
| Y: K, r | +** | 0.63 ** | - | 0.0105 ** (λK) |
| Y:Research and Development(Gov), r | +** | 0.64 ** | -0.218 ** | 0.0579 ** (λRD(G)) |
| Y:Research and Development(Pvt), r | +** | 0.61 ** | -0.326 ** | 0.0579 ** (λRD(P)) |
In this formulation, net output is positively associated with the respective input prices in all cases. That is, a rise in real prices of each input is associated with a positive change in year-to-year net output. In the case of the real wage, the rise in wage costs would indicate labour shedding and other things being equal a fall in production. The data indicates, on the other hand, that production is maintained possibly through other input adjustments to real wages. In the case of real interest rates, the rise in costs should indicate capital stock adjustment at the margin and hence less capital and less production. The data indicates that production is maintained possibly due to the smallness of the change in capital stock and also lags in response.
The adjustment parameters are relatively high compared with similar functions in Table 1. Some 61-76 % of theoretical response to price and input volume changes is achieved in one year on average.
The Research and Development response coefficent is negative in both cases formulated and suggest output changes are inverse to the build up of R&D capital at the beginning of the production year. As previously suggested, the perpetual inventory method does not make a particularly good measure of the "technology capital" available (Johnson 2000). The sign of this coefficient does not prevent the estimation of the respective technological growth rates.
Technological change in the labour and capital inputs follows previous estimates with labour augmenting growth dominating. In the Research and Development equations, the efficiency growth of productivity appears to be considerably higher whether it is government or privately funded, and suggests a far greater shift in the R&D isoquant than the other two inputs. Thus going by these estimates, technology is still labour augmenting [as compared with capital] but R&D capital input is also strongly augmented by technology. The isoquant for R&D is bent upwards as compared with capital. By implication the isoquant for labour and R&D capital is symmetrical to the origin though we don't know what the elasticity of substitution is.
There are limits to the usefulness of aggregate time series data for production function analysis. In this data set there is not sufficient range in the price and quantity year-to-year changes to allow measurement of the parameters desired. If the capital/labour ratio had varied more, some sensitivity to prices could be expected, and better estimates of the elasticity of substitution derived. This follows from the recognised relationship between the annual investment component of capital stock and yearly variations in income and the price of capital.
It is also clear that the aggregates conceal the individual plans and actions of firms in an industry. Wider price responsiveness in some firms may be cancelled out by the actions of others. The index numbers representing volume changes in the main production variables may also be biassed by base year problems in their construction. It may be that index numbers derived from Fisher indexes would be be more accurate of producer intentions.
There is good evidence for labour augmenting technical change in the agricultural sector. Technical change enhances labour productivity more than capital productivity even though many techniques require capital investment. The role of intermediate inputs require further investigation as they are clearly part of the same production process.
The response to capital investment in Research and Development remains disappointing. Better specified production models do not elucidate the role of R&D any more clearly than previously. It remains a problem how to specify the measurement of a capital stock of ideas built on the cost of R&D investigation. Has anyone a better methodology?
Intriligator, M.D. (1978), Econometric Models, Techniques, and Applications, Prentice Hall, New Jersey.
Johnson, R.W.M. (1971), A Note on The Proportional Elasticity of Substitution Production Function, NZ Economic Papers, 120-125.
Johnson, R.W.M. (1972), Efficiency Growth in NZ Agriculture: A Review, Economic Record 48 (2), 76-91.
Johnson, R.W.M. (1999), The Rate of Return to New Zealand Research and Development Investment, Paper presented to NZ Association of Economists Conference, Rotorua.
Johnson, R.W.M. (2000), Methodologies for Measuring the Accumulated Knowledge Base in Research and Development, Paper presented to NZ Association of Economists Conference, Wellington.
Mukerji, V. (1963), Generalised S.M.A.C. function with Constant Ratios of Elasticities of Substitution, Review of Economic Studies 32(3).
Philpott, B.P.(1994); Data Base of Nominal and Real Output, Labour and Capital Employed by SNA Industry Group 1960-1990, RPEP Paper 265, Victoria University.
Philpott, B.P. (1999), Provisional Estimates for 1990-1998 of Output Labour & Capital Employed by SNA Group, RPEP Paper 293, Victoria University.
Solow, R.M. (1967), "Recent Developments in the Theory of Production, in The Theory and Empirical Analysis of Production, (ed: M.Brown), National Bureau of Economic Research.